
Note that the Jaccard distance measure derived using the cardinality operator \(\mu (A \cap B)\)  as defined by Equations (REF ) and (REF ) is symmetric and follows the identity of indiscernibles, and thus may be considered to be a semi-metric. The triangle inequality holds in the case of binary equivalence [1]} but not generally for soft equivalence relationships. Equation (REF ) may be used to define the cardinality of a single set \(A\)  as \(\mu (A)=\mu (A \cap A)=|A|\)  both for hard and soft equivalence since, since for a given element \(f_i \in A\) , \(\max _{f_j \in A}\mathcal {K}(f_i,f_j) = \mathcal {K}(f_i,f_i) = 1\) . The standard Jaccard index in Equation (REF ) is thus a special case of the soft Jaccard index in the case of a binary-valued kernel \(\mathcal {K}(f_i,f_j) \in \lbrace 0,1\rbrace \) .
